Lemma 4.8.1. Let \mathcal{C} be a category. Let F, G, H : \mathcal{C}^{opp} \to \textit{Sets} be functors. Let a : F \to G and b : H \to G be transformations of functors. Then the fibre product F \times _{a, G, b} H in the category \textit{PSh}(\mathcal{C}) exists and is given by the formula
(F \times _{a, G, b} H)(X) = F(X) \times _{a_ X, G(X), b_ X} H(X)
for any object X of \mathcal{C}.
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